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Prof, G. Hinrichs on Planetology. 






03 



I'rt'f 



P^OM THE AMEBICAX JorE^'AL OF SCIEXCEj VOL. XXXIX.] 



INTEODUCTIO^^ 



MATHEMATICAL PRIKCIPLES OF THE 



NEBULAR THEORY. OR PLAXETOLOGY 



BY GUSTAYUS HlN'RICHS, 

Professor of Physics and Chemistry, Iowa State University. 



The nebular hypothesis — the boldest thought that ever elevated 

the human mind, by bringing us, as it were, in sight of the mys- 
terious fiat of the Almighty — was, in its great general features; 
unfolded almost at the same time by Germany's deepest thinker, 
the Konigsberg philosopher, Immai^uel KatsT, and by Pieeke 
SiMo:s" BE Laplace, the greatest mathematiciaa of France. It 
is truly the closing stone in the philosophy of the celestial 
vault; for Copernicus and Kepler made us behold the founda- 
tion. — the first, by placing the sun as the lantern of the world in 
the center, and surroundii^ig it with the planets — the second, by 
destroying the cycles and unravelling the harmony of the spheres 
in his immortal laws; and after the existing phenomena had 
thus been rightly viewed, Newton made us behold the invisible 
bond that connects the members af the systesn, while at length 
Kant and Laplace pointed out t-o us the hand that at '''the 
beginning" projected these celestial balls into space and there- 
by insured the continued existence of the system. 

But notwithstanding this noble parentage and i's ■:T:::g :::e 
logical sequence of the discoveries in the theory of o:sr:::s L:aie 
by Copernicus, Kepler and Newton, the nebular theory ei :}5 
as yet but slight consideration among astronomers. Arago is 
the only one of these who has deigned to consider it earr.esi'y, 
and he probably did so more in his capacity as a physicist ihan 
as an astronomer. 

' Arago^ Astronomie Populaire, ii. 7. Paris and Leipsic, 1855, 



G. Hinrichs on Planetology. 3 

The reason of this neglect seems to be the incomplete state in 
which even Laphice himself left the theory. Direct observation, 
moreover, seemed to contradict some laws given as necessary 
consequences of this hypothesis. 

We have already, in a former article,'^ tried to vindicate the 
theory in this last respect by showing that the hypothesis is 
really confirmed even in these apparently contradicting observa- 
tions. We will now endeavor to give a somewhat more com- 
plete development to the fundamental principles of Kant and 
Laplace, and to exhibit the exact position of the nebular theory 
itself, hoping thereby to show that this theory, if we only study 
it earnestly and patiently both by experiment and analysis, fully 
deserves our confidence. 

As this subject is as vast as it is difficult, we beg the critic 
alvva37s to keep in mind that we do not pretend to give a treat- 
ise, but merely offer an introduction to this almost new field of 
analysis. 

We commence with a short survey of the fundamental prin- 
ciples and the aim of the theory of the solar system, in order 
clearly to understand why the nebular theory is necessary, what 
it will have to accomplish, and how far it already has done its 
duty. 

§ 1. The fundamental constants of the Solar system. 

As the discovery of a law of nature is but the reduction of 
the infinitude of observed quantities to a few constants by means 
of a function, the algebraic expression of the law — we see that 
the progress of astronomy to a great extent must be identical with the 
reduction, of the number of such constants. This is fully borne 
out by the history of the science. For, while the Ptolemaic the- 
ory^ of the planetary motion required the radii and inclinations 
of seventeen different circles to express the observed motions of 
Saturn, Kepler reduced this number of constants to three, the 
semi-major axis, eccentricity, and inclination of the orbit. This 
very principle is also placed by Laplace* at the head of his Me- 
canique Celeste. 

We may therefore trust in this principle, and with Laplace 
try to reduce the number of indispensable constants. We must 
first, however, ascertain which are those constants that are now 
considered fundamental or indispensable. 

The constants of the solar system now exclusively deduced from 
observation are : 

1. The 7nass, m^ of the planet. 

* The density, rotation, and relative age of the planets : this Journal, Jan., 1864. 
' Fracastor; see Bailly, Histoire de I'Astronomie moderne ; vol. i. Paris, 1779. 

Eclaircissements, livre iv, § 23, and livre viii, § 27. 

* II importe extrememeut d'en bannir tout empirisme et de la r6duire d n'em- 
prunter de I'observation que les donnees indispensables. — Mec. Cel. — Flan. 



4 G. Hinrichs on Planetology, 

2. The figure. On assuming a primitive fluidity and a small 
velocity of rotation, theory gives an ellipsoidal figure of the 
planets; hence not independent of observation. 

8. The ellip/icitt/ of the planet. Theory may assign to it a 
higher and a lower limit by means of the mass (1) and the an- 
gular velocity (6) — but though the connection of this constant 
with others thereby is manifest, still its exact value can only be 
derived from observation. 

4. The volump^ or diameter^ of a planet. Combining this con- 
stant with the first (mass) we obtain the density. 

5. The plane (or incHfiation of the axis), 

6. The direction^ and 

7. The velocity of rotation. Though this last element bears 
relation to others, still its exact value can only be obtained from 
observation. Even if Kirkwood's law of rotation^ should prove 
to be perfectly exact, this element would continue to be a funda- 
mental constant as lon^ as that law remains an empirical one. 

8. The distance^ a, of a planet from the sun, or the semi-major 
axis of its orbit. By the theoretically proved third law of Kep- 
ler, we get from this constant the periodic time, T, requiring only 
the conistant, in, of gravitation to be known, and this latter is the 
same for all planets. 

9. The plane or inclination, i, of the orbit. 

10. The direction of the motion. The velocity is given by the 
distance. 

11. The eccentricity of the orbit. This constant is fundamental, 
for the theory of gravitation only proves the orbit to be a coiiic 
section of some kind. The eccentricity can only be found from 
observation. 

12. The numher of satellites of a planet is also fundamental — 
and for each of them the same eleven constants have to be taken 
from observation; the first seven even are required by the sun. 

From these twelve empirical data, theoretical astronomy can 
deduce the motion of the corresponding planet. The whole 
number of empirical constants is not at all inconsiderable ; for 
8 principal planets, constants 1-1 2, - - - 96 
80 small planets, " 8-11, - - - 320 

23 secondary planets, " 1-11, - - - 253 
The sun, " 1-7, ... 7 

Total number of constants, 6Y6 

to which the corresponding constants for the comets would have 
to be added. 

What, in the face of this great number of constants that as- 
tronomy has to borrow from observation, shall we say about the 
boasted perfection of this science ? Is it not in science, as in 

• This Journal, ix, 895, May, 1850; also xiv, 210, Sept., 1862. 



G. Hinrichs on Planetoiogy. 5 

morals, that self-adoration hinders progress ? Can any astron- 
omer who has not merely studied the details of the celestial 
mechanics, but also kept in mind the great principle laid down 
by its author in his "Plan" — can he still pretend that Newton's 
theory of the solar system merely needs further development, 
seeing that the few bodies of this system require him to borrow 
about seven hundred constants from observation? 

We shall honor the memory of Kewton much more by trying 
to go beyond the results of his labors than by stupidly worship- 
ping" the same, and thus arresting the progress of that science to 
promote which he spent his life. 

§ 2. These fundamental constants sustain remarkable relations to each 

other. 

The jSTewtonian theory of gravitation simply accepts these 
constants as observation gives them. For if our earth had Ju- 
piter's mass, the rings of Saturn, the moons of Uranus and its 
axis in the ecliptic, the latter perpendicular to the orbit of Ju- 
piter, a retrograde motion in a hyperbolic orbit — it still would 
as fully and as beautifully confirm the theory of universal grav- 
itation as it does now ; for, let us openly and frankly acknowl- 
edge it, these constants are independent of the theory of gravi- 
tation because the latter is independent of the former. 

But, though this theory does not give any reasons for any kind 
of dependence between the often mentioned constants, observa- 
tion shows that they sustain very remarkable relations to each 
other y or in other words, there are relations and laws in our solar 

" The literature of astronomy toems with implicit instances hereof; but we find 
also direct expressions of this feelins:, like the following : 

Enfin, nous avons vu que ces resultats eux rnemes peuvent se composer en un 
seul et se representer par une lol unique, celle de la Pesanteur universelie ; parve- 
nus a principe nous nous voyons en quelque sorte eleve a la source commune de tons 
lesfaits aslronomiques ; tous en derivent de la maniere la plus simple et ils y sont 
en quelque sorte comme concentres. Nous avons done pour ainsi dire decompose 
le systeme du monde, nous Vavons reduit a son element unique, et nous I'avons en- 
suite recompose — Biot, Traite elem. d'astronomie physique. Paris, 1805. Conclud- 
ing remark of the work. 

This "element unique" is rather singularly unique, requiring no less than seven 
hundred elements to be borrowed from observation alone ! 

' Newton was aware of this — indeed, nobody can help seeing some of these re- 
lations. In the scholium to the third book of Principia he says: 

" Planetae sex principalis revolvuntur circum solera in circulis soli concentricis, 
eadem raotus directione, in eodem piano quamproxime. Lunse decern revolvuntur 
circum terram, jovem et saturnam in circulis concentricis, eadem motus directione, 
in planis orbium planetarum quamproximos. Et hi omnes motics regidares originem 
nan habent ex causis m.echanicis." Edit, le Seur et .Jacquier Geneva 1749-42. 

It is customary to censure Kepler's fancy in contrast to the solidity of all New- 
ton's words; still a sentence like the above is much more objectionable in science 
than the boldest fancy, for the latter is not accepted without severe scrutiny, while 
the former is repeated as a sacred truth. If Newton had written "mechanical 
causes known to me" instead of by "mechanical causes," simply to imply that he 
knew them all, he would have prevented many a drawback that has encumbered 

Am. Jour. Sci.— Secoxd Series, Vol. XXXIX, No. 115.— Jast., 1865. 
7 



6 G. Hinrichs on Planeiology. 

ivorld of a still higher order than those deduced from graviiation. 
Thus, the inclinations of the orbits of the principal planets, in- 
stead of being uniformly distributed over the first quadrant, are 
all very small ; and their direction, instead of being as often 
retrograde as direct, are for all planets and most satellites direct. 
Instead of having the eccentricities regularly varying from to 
00 , we find them for all planets nearly zero ! The same may be 
said of any of the above fundamental constants, and not least 
of the distance, as it is found approximating to Titius-Bode's 
law, that is, to 

0„=4 + 3-2"'i (1) 

But quantities that sustain mutual relations to each other are 
the particular values of a certain function for definite given 
values of the variable quantities ; hence, if we intend to be true 
to the spirit expressed by the words of Laplace above quoted, 
it is a problem legitimately belonging to astronomy to find these 
functions of which the fundamental constants are hut particular 
values. 

Let us boldly face this great problem and not desist though 
astronomers tell us that it is not part of their science. 

§3. The fundamental constants satisfy/ ike conditions of stahility of the 

system. 

Since relations exist between the values of the fundamental 
constants, we may ask for the most general expression of these 
relations. The principal of these relations are, by Lagrange and 
Laplace, proved to be such as to fulfill the conditions of stability 
of the system. These conditions are — 

1. Incommensurability of the times of rotation, ensured by the 
distances forming an exponential series (1). 

2. The central mass vastly preponderating, and the greatest 
masses revolving where the mutual distances are the most con- 
siderable. 

3. The direction of all motions is the same. 

4. The plane of all orbits is and remains nearly the same, because 

2 m/\fa.tg^iz=zc^, (2) 

is and remains a small quantity (the letters having the same sig- 
nification as in § 1). 

science. Less arrogant, because more true, he is when writing to Burnett (about 
1680-81), " but yet I must confess I know no sufficient cause of y® earth's diurnal 
motion." 

Even Biot, notwithstanding his blind admiration, above cited, cannot help seeing 
something more than gravitation can account for, when noticing the harmony in 
the rotary and translatory motions. He says: " cet accord, qui tieiit sans doute au 
premieres causes qui ont determine les rnouvemerits planetaires, est un des phenome- 
nes les plus remarquables du systeme du moncfc. — Astron. physique, vol. iv, chap, v, 
p. 467. 



G. Hinrichs on Planetology . 7 

5. The eccentricity^ e, of the orbit is and remains nearly the same, 
because 

^ m\/a.e^ z=C2i (3) 

is and remains a small quantity. 

6. The density, d, is such that the diameter of the bodies is 
small in comparison to their distances. 

We may add — 

7. The Jonn of the planets is sucb that the influence of the 
deviation from a sphere is the smallest possible. 

§ 4. Gravitation is insufficient. 

The laws of Kepler are grand — as ^7elI as ISTewton's theory 
in accounting for them ; but the above laws of Lagrange and 
Laplace are certainly of a superior order, and the theory of 
gravitation in failing to give even a shadow of a reason for these 
laws proves itself to be not the whole truth : we must go be- 
yond this force ! 

Astronomers seem to forget the history of their own science; 
for how could they otherwise deny the legitimacy of accounting 
for the fact that the above laws express the stability of the 
world? Had not astronomers at the time of Kepler the same 
reason to be satisfied with his laws as astronomers have now in 
abiding by the laws of stability ? And is it not as urgent to 
discover the causal connection between these laws ensuring 
great duration to our system, as it was to find in gravitation the 
mechanical cause of those laws ruling the spheres at the time 
being? 

Kow the hypothesis of Kant and Laplace will be found to 
account for the laws of stability as rigidly as the hypothesis of 
Newton accounts for the laws of Kepler ; why, then, deride the 
former and adore the latter hypothesis? Or do we even forget 
that "the principle of gravitation" is but a ^^ hypothesis T'' Did 
not Newton himself consider it as such? Is not this force fully 
as mysterious and fully as much beyond the reach of direct ob- 
servation as the chaos of Kant and Laplace? The former we 
assume as continually acting, because we find that the motions 
are such as this force would produce {provided a tangential force, 
of which gravitation knows nothing, also acts in a definite man- 
ner, etc., etc.). Why not also assume the latter as having been 
real, if we find by the same mechanical deductions that the 
existing harmony, as expressed in the stability of the system and 
ensured in the mutual relations of the fundamental constants 
follows directly from the above-named chaos? If in the one 
case we reason from fact or law to cause — why not also in the 
other, provided our conclusion is as legitimate? 

Here astronomers will not fail to object that this last condition 



8 G. Hinrichs on Planetology. 

is not satisfied. We fully admit this ; but beg them, to remember 
that it has taken two centuries of labor to ensure this legitimacy 
to gravitation — that Newton did not leave gravitation as a mere 
suggestion (as such it had existence before him), but in his im- 
mortal Principia gave the necessary mechanical firmness to this 
hypothesis : how different has been the lot of Kant and La- 
place's hypothesis! The first of these expounded it rather fan- 
cifully in connection with speculations on the inhabitants of dis- 
tant globes;^ the latter only gave a few bold and deep outlines 
of the nucleus of the theory! What would to-day be the esti- 
mation of gravitation if we, instead of New^ ton's Principia^ only 
had a few of Hooke's sublime guesses, if these had only been 
considered by men like Fontenelle' instead of being investigated 
by Euler, the Bernoullis, Lagrange, Laplace, Gauss, Hansen, 
Plana, etc. ? 

Can anything be more unjust than exaltation of the hypothe- 
sis of Newton — this deservedly cherished subject of the master 
minds of two centuries — above the hj^pothesis of Kant-Laplace, 
which, being too early left even by its astronomical parent, has 
been ever since considered an outcast in the world, endangering 
the reputation of any one who would dare to touch it? 

We will adopt this almost forlorn hypothesis as a mere hypoth- 
esis — we will patiently and carefully trace its bearings by means 
of asvrigid an analj^sis as we can command in this most intricate 
field ; we will minutely compare the results thus obtained with 
the actually observed state of things; and if we find the corres- 
pondence heiween idea and phenomenon, between analysis and ob- 
servation, to be very close, we hope that those who have analysis 
more at their command than v/e, will pay as much attention to 
this high branch of astronomy as has been, and deservedly con- 
tinues to be, bestowed on Newton's hypothesis of gravitation. 
If our feeble endeavors only succeed in making Kant-Laplace's 
hypothesis admitted as such among analysts, we shall have accom- 
plished ail we desire: for then this hypothesis will soon be con- 
sidered as firmly established a principle as gravitation," or as 
the fact of the rotation of the earth, ^' which latter — notwith- 
standing Foucault^s pendulum and gyroscopes — still remains un- 
proved hy ocular evidence^ all "demonstrations" being in fact 

^ AUgemeine N'aturgeschichte und Theorie des Himmels, 1755. 

^ Theorie des Tourbillons Cartesiens avec des reflections sur Tattraction (1752). 
Also his Entretiens sur la pluralite des mondes. 

" Gravitation was always treated as a mere hypothesis by the Cartesians ; the 
work of Fontenelle above cited offers the instance most generally known. 

^^ The scientific prejudice existing against the nebular theory is perhaps as inju- 
rious as the religious prejudice once " r enisling" the motion of the earth. See the 
amusing statement in the preface to the 'Dei massimi sistemi,' (ed. Padova, 1744,) 
that the " ' Moto della Teria' non puo ne dee amettersi se non come pura ipotesi 
matematica. che serve a spiegare piu agevolmente certi fenomeni." Also Galileo 
liimself, in his Dialogue {Opere compl. Firenze. Vol. i, 1848, pp. 387 and 447). 



G. Hinrichs on Planetology. 9 

inductive, reasonings with our senses. Thus, in the latter, we 
see the plane of the oscillations rotate, but conclude it to be the 
earth. It requires at least as much mental effort to apprehend 
its true bearing as the simple reference to the diurnal motion of 
sun, moon, and stars. 

§ 5. How far the nebular theory accounts for the stability of the system. 

The theory of gravitation can never, therefore, account for 
the stability of the solar system. How far, then, does the nebu- 
lar theory explain those great fundamental conditions of the 
system that ensure not only the harmony of the solar world but 
even make this harmony (almost) permanent? 

In order to invite physicists and astronomers to the perusal of 
the following introduction, we will try to give a simple answer 
to this most important question. 

I. The plane of all planetary orbits must he nearly the same (see 
§1, 9, and §3, 4, also § 10). 

This theorem has been clearly seen by Kant and Laplace ; it 
is the most immediate expression of the hypothesis. Mr. Trow- 
bridge has recently pointed out^^ to us a very interesting conse- 
quence hereof, viz : the most distant planet must move in the in- 
variable plane ; and, indeed, the inclination of the orbit of Nep- 
tune is 1° 47', that of the invariable plane 1° 41'. 

II. The direction of all planetary revolutions must be the same ; 
this obvious consequence of the hypothesis accounts for § 3, 3, 
and §1, 10. 

III. The eccentricity of the planetary orbits must be very small — 
accounting for observation, § 1, 11, and condition of stability, 
§3, 5 and §10. 

This proposition has been deduced in general reasonings by 
Kant and Laplace; in the following we will try to give a dem- 
onstration of it. 

lY. The planetary distances are such that the successive planets 
were evolved at equal intervals of time ; or if ^=1, 2, 3, ... . re- 
spectively for Mercury, Yenus, Earth . . . . , then the distance is 

a,=:«-l-^.y^ (4) 

where ■«, i?, / are constants, and t the age of the planets above that 
of Mercury. From this follows the condition of stability that 
the periodic times are incommensurable (§ 3, 1). Besides, it is seen 
that this law accounts for the empirical law of Bode (§ 1, 8). 

The analytical demonstration of this law is one of the princi- 
pal objects attempted in the present introduction. (See § 13.) 

Y. The mass of the more distant planets is the greater^ on account 
of the greater space from which the material of the planet was 
condensed (space increasing according to lY). This is confirmed 

" On the Nebular Hypothesis, g 24; this Journal, November, 1864, page 355. 



10 G, Hinrichs on Planetology. 

by tke fact that the sum of the four great exterior planets is 480 
times the mass of the earth, while the sum of the masses of the 
four interior planets is but twice the mass of the earth. Further- 
more, the mass must increase toward the sun — as the density from 
which the rings were formed was greater nearer the center. 
This is confirmed, as the mass of Neptune is 25'6, the mass of 
Uranus but 14*5, so that the mean of the two most distant is 
20; the mass of Saturn is five times as great (lOl'G), and the 
mass of Jupiter, again, three times greater (339*2). The mini- 
mum in the case of the mass of Uranus is evidently produced 
by the simultaneous influence of both the above principles. 

The mass of Jupiter is a maximum ; it is so great that the next 
following ring was broken up into fragments hy the perturbing influ- 
ence of so stupendous a mass — thus originating the host of aster- 
oids, and perhaps also the meteorites. 

The very small mass of the interior planets as compared with 
the exterior ones is not astonishing, if we remember that the 
inter-planetary space between Jupiter and Saturn is to that be- 
tween Venus and the Earth, as 10 ^^ -5-23 to l-00='--72=', or as 
860 to '63, or nearly as 1300 to 1. The mass of Jupiter is to 
the mass of our earth as 340 : 1, thus giving us still some mar- 
gin for an increase in density toward the center of the nebula. 

Being as yet unable to give the precise theoretical law of the 
masses, we are obliged to make the above few suggestions, in 
order to show that the nebular hypothesis at least gives a gene- 
ral law of distribution of the planetary masses in conformity 
with § 1, 1, and § 3, 2. (Prof Kirkwood takes a similar view 
of the asteroids; see this Journal, 1852, xiv, 214.) 

YI. The figure of the planets (being a condensed vapor) must be 
an oblate spheroid of 

YII. Small ellipticity, because 

YIII. The velocity of rotation is but small (compare § 1, 2, 3, 7, 
and §3, 7). 

This last proposition is based upon the fact that the moment 
of rotation is but the difference between the moment of revolu- 
tion of the exterior and interior part of the planetary ring. 

Still, the exact amount of this velocity, as well as the period of 
rotation of the sun, has not yet been deduced from the nebular 
hypothesis; we have often attempted it, but as yet have not 
been able to solve this difficult problem. 

Prof Kirkwood has found '^ the empirical law of the velocity 
of rotation, a law analogous to the third law of Kepler. We have 
repeatedly arrived at expressions similar to (but not identical 
with) Kirkwood's law. 

IX. The Plane; and 

^^ As above, this Journal, 1850, ix, 395. 



G. Hinriclis on Planetology. 11 

X. The direction of rotation of the planets has been considered 
at variance with the nebular theory, ever since the discovery of 
the lunar svstem of Uranus. We believe that our analysis of 
this problem'^ shows that the rotation of Uranus and IS'eptune, 
both as to position of the axis and direction of motion affords a 
very interesting confirmation of the theor3\ See § 1, 5, 6. 

XL The density of the planets has also been considered as 
being adverse to the theory; but if, as necessary, the influence 
of the age is taken into account, it is found that the minimum 
density exhibited by Saturn is demanded by the theory.'' Com- 
pare §1, 4, and § 3,*' 6. 

XII. The number of satellites was already shown, by Kant, to 
increase with the distance from the center of the nebula. Though 
not usually given as such, it nevertheless is a condition of sta- 
bility of the svstem — at any rate it is conformable to observa- 
tion"'(§l, 12)." 

The rings of Saturn are best considered as a host of satelloids, 
corresponding to the planetoids (and meteorites) of the solar 
world — thereby accounting for the excessive thinness and the 
subdivisions of the rins^s.'^ 

CD 

In looking back upon the preceding account of the present 
aspect of the nebular theory, it will be seen 

A. That the four great fundamental conditions of stability re- 
ferring to the system at large are now satisfactorily deduced from 
the hypothesis of Kant-Laplace (I-IY above). 

B. That the problem of the mass (Y) and the number of satel- 
lites (XII), though not completely evolved, still is sufficiently 
comprehended to enable us to say that the analytical solution is 
possible; and 

C. That the elements referring to the single planets^ or rather 
their subordinate systems, are, with the exception of the exact 
law of rotation (VIII), fully deducible from the fundamental 
hypothesis of Kant and Laplace. 

We see, then, that the fundamental constants of the solar sys- 
tem, which number about seven hundred (§1), exhibit very re- 
markable mutual dependencies (§ 2), which are such as ensure 
the permanence or stability of the system (§ 3), which Newton's 
law of gravitation cannot account for (§4). Though they offer 
a higher problem for theory than Kepler's laws, astronomers 
have hitherto been unwilling to recognize the analysis of the 
above conditions of stability as part of their science. Laplace, 
while instrumental in bringing to light the great laws of the 
stability of the system, independently reproduced the bold hy- 

" On the density, rotation, and relative age of the planets; this Journal, 1864, 
xxxvii, 36, 48. 

" As above, p. 49. 

^® On the densitv, etc.; this Journal, 1864. xxxvii, 54. 



12 G. Hinrichs on Planetology. 

pothesis of Kant, and tbough this has been most grievously 
neglected by analysts and astronomers, still it now affords us a 
full solution of the four great harmonies ensuring the perma- 
nency of the solar world, and also solves most, and at least in- 
dicates the solution of, all other problems relating to the har- 
mony of the fundamental constants of the solar system. 

May we not hope that astronomers will begin to bestow on 
this theory some share of their labor ? 

§ 6. The Hypothesis. 

We assume, with Kant and Laplace, as the primitive condi- 
tion of the solar system, or as 7iehula: 

The space of the solar system was filled with matter having a mo- 
ment of rotation. 

This matter is endowed with the* same forces we know it to 
possess; a simple calculation will furthermore show that it was 
a highly rare vapor. Its chemical constitution we will leave out 
of consideration for the present ; we therefore consider it as com- 
posed of the elements we know here on earth, many of which 
we now know to be found on the sun, and are probably also on 
the distant stars ;'^ still there can be no doubt but that many 
more elements exist than we are acquainted with. Many of the 
spectral lines even of our own central star are irreducible to 
spectra of known elements. We therefore mean simply to say 
that at the above primitive period the elements had been created. 
I hope at some future time to publish an attempt at a mechanical 
theory of the elementary bodies^ which has occupied my time for 
about ten years, and wherein I endeavor to show the physical 
properties of the known elem^ents to be definite functions of their 
atomic number and form. Accordingly, there would yet remain 
a more primitive condition, the existence of the one primitive 
matter (IJrstoff) which would be considered as the direct crea- 
tion of the one God. 

The rotation of the nebula is not to be thought regular, but 
simply amounting to a certain momentum. I have elsewhere^* 
tried to show that such rotation may be considered as the effect 
of a difference of any kind between the primitive forces of attraction 
and repulsion wherewith we know matter to have been endowed. 

If, therefore, these views should be well founded, we should 
have arrived at the grandest principle we can conceive of in the 
present state of our knowledge; we should be able to see how 
from created matter alone the whole of the solar system has been de- 
veloped; we would be enabled to conceive the almighty ^a^ as 

^^ Rutherford, Astronomical Observations with the Spectroscope; this Journal, 
1863, XXXV, 71. Above all, Bunsen's and Kirchhoff's memoirs on their great dis- 
covery. 

" This Journal, 1864, xxxvii, 52. 



G. Hinrichs on Planetology. 



13 



one single act. How mucli such a theory would tend to elevate 
our conceptions of the great Author, we cannot here develop. 

In the present paper I shall not go farther back in time than 
to the existence of the nebula of Kant and Laplace as above 
defined." 

§ 7. Plateau'' s experiment. 

Before entering upon the analysis of the nebula, we must refer 
to the experimental evidence of the nebular theory afforded by 
the beautiful experiments of Plateau, detailed in his Memoire 
sur les phenomenes que presente une masse liquide lihre et soustraite 
a Vaction de la pesanteur, Pt. I (Neuv. mem. de I'Acad. de Brux- 
elles, vol. xvi, 1843). His results are: 

1. A liquid, subject only to the action of its molecular forces 
assumes the form of a perfect sphere (§ 2). 

2. This globe is flattened at its poles, if subject to rotation, 

" It is perhaps not out of place here to give a synopsis of the different distinct 
ages that are characterized by a further individualization or a new direct creation 
according to the views above indicated. 

Three (or four) direct acts of the Deity may be recognized, viz : the creation of 
matter, of life, of mind (and the redemption). The formation of the elements out of 
matter characterizes the first age ; the formation of the solar world, with its plan- 
ets, moons and central sun, the second age ; while the third age beheld the develop- 
ment of our earth from a vaporous ball to its present shape ; in the fourth age, life 
was created in the form of plards and animals ; in the ffth age, mind, the investi- 
gating mind, was introduced by the creation of man, the cephalized animal; while 
a sixth age will behold the destruction of the whole system, occasioned by the ex- 
Unction of the solar body and the resistance of ether. (See this Journal, 1864, 
xxxvii^ 56.) 

To every age correspond two sciences : the first relates to the development — 
Whewell would call it the Faleticlogy of the age — while the second relates to the 
actually existing product of the development or creation of that age, i. e., the sci- 
ence. Thus W8 obtain the following general view of the natural sciences: 



Agej I 1 2 1 3 1 


4 

11. Cre 

ation of 

Life. 

Paleon- 
tology. 

Botany, 
Zoology 


5 

lll.Crt.- 
ation of 

Mi7id. 

Arche- 
ology. 

History. 


(6) 




1. Cre 
De 
Elements. 


ation of Mat 
velopment o 
Solar system. 


TEE. 

f 
Earth. 


Destruc- 
tion 
of the 
world. 


Paletiology 


Atomology 


Planetology. 


Geology 


Science. 


Physics, 
Chemistry. 


Astronomy. 


Geog- 
raphy. 



These names have of course to be taken in their widest sense: thus geography 
stands for physical geography, meteorology, etc., and history comprehends not only 
political but also the intellectual history- of the human race, thus including again 
all the sciences in their historical development. We see how " planetology" is 
allied to geology and astronomy. 

Am. JonR. ScT.— Second Series, Vol. XXXIX, No. 116.— March, 1865. 



14 G. Hinrichs on Planetology, 

8. If the rotation becomes sufficiently rapid, the wliole glohe is 
transformed into a ring in the equatorial plane (§§ 11-14) which, 
by continuing the rapid rotation of the large central disc, even 
loses its regularity, and separates into small masses which imme- 
diately assume globular forms. "But," continues he (§19), 
" this is not all ; one or more of tliese spheres are always seen to 
take, at the instant of formation, a motion of rotation on their 
own axes, and this motion is almost always in the same direc- 
tion as that of the ring."" He even found that still smaller 
globes were formed. 

4. If a small disc is put into very rapid rotation, a ring is 
formed, while a part of the original globe remains on the axis 
(§21), so that Plateau rightlj^ concludes with sajing (§27) that 
most of the phenomena of the relative configuration of the 
heavenly bodies have been reproduced by him on a small scale." 

$ 8. The condition of the primitive Nebula. 

Whatever may be the distribution of matter in the nebula, 
and however the particles may move, the beautiful theorem — 
the magna charta of the nebular theory — obtains, i. e. the nebula 
possesses the invariable plane of maximum areas, or if &> represents 
the angular velocity, r the projection on the invariable plane of 
the radius vector drawn out from the center of gravity, m the 
mass, and A the projection of the area swept over by r in the 
unit of time, we have, C being a constant, 

2mA=C] .... (5) 
or, as the centrifugal force y of the particle m with respect to the 
principal axis is 

rinwSr (6) 

while, at least for a very small unit of time, 

A = ^a.r2, . . . . (Y) 

we have also 2mr^y^ = 2C. ... (8) 

On account of the resistance of the ether, C will not be quite 
-constant, but decrease in time; still it is apparent that, as a first 
approximation, we may neglect this resistance by considering C 
strictly constant. 

Now, by the mutual attraction of the particles, the nebula is 
<;ontinually becoming more dense, or r is continually decreasing; 
hence, by (8), the centrifugal force of any particle in the nebula is 
continually increasing. 

^° Of course ; for in this oil the density is the same. See our article, this Jour- 
■nal. 18^4, xxxvii. 51, 6 = 0. 

^^ Malgre la difiference des lois que suivent les forces attractives dans ce cas et 
celui des grandes masses planetaires, nous avons vu se produire, en petit, une re- 
presentation frappante de !a pluspart des phenomenes de configuration relatifs aux 
corps celestes. 



G. Hinrichs on Planetology, 15 

But the force of gravity at the surface is likewise constantly 
increasing; for we may without materially erring conceive the 
mass below the particle to remain constant, but then gravity is 
inversely as the square of the radius, or ra/pidly increasing with 
the progress of condensation . 

But these two forces determine the figure of the Nebula, However 
irregular the figure may be at first, we see that the moulding 
forces, by constantly increasing, will at length shape the nebula 
accordingly. From Plateau's Experiments (see above, § 6, re- 
sult 1, 2) we know this shape to be a flat ellipsoid. Laplace^ 
has demonstrated that but one sirigle oblate ellipsoid of revolution 
will be produced by these forces, i. e. 

z2^ni{x2-{-y2>^=:a2 , ... (9) 

the plane x, ?/, coinciding with the invariable plane, being the 
equator, z the axis of rotation = 2a , and 

m — -yi=, Izzztyd, Bin = 6, . (10) 

e being the eccentricity of the meridian ; hence the equatorial 
semi-axis 

= ^X1+^- • • • (11) 
Assuming, for a moment, the nebula to be homogeneous, w^e 
can determine the eccentricity by the density d, and the angidar 
velocity w (j, e. by (6) proportional to the square root of the 
centrifugal force ^ at a unit of distance). Laplace found, if the 
mass of the whole nebula be M, and its moment of inertia E, 
taiat 

E=i^^n8a^{\-\-l^)^gz=z'^^a^U', , . (12) 

o 

U = inSa,^\+X2) = ind -^^- . . (13) 

Of; ■R'2 1 
,'=^(4.^)^; (14) 

,, ,, 9X + 2g'A3(i+A2)-f 

arc (to = A) m ! — i— — A— 1 '- — ; . . (15) 

V y / 9-1-3^2 ^ ^ 

which last equation he shows to have but one single positive real 
root, so that A or e has but one value, if g, the centrifugal force, 
and (J, the density, are given. 

But the density is certainly not constant throughout the whole 
nebula ; but as this nebula is a gaseous body, S will be deter- 
mined by y and gravity and, just as in the case of our earth, be- 

* M(?icanique Celeste, Li v. iii, chap. Ill, § 21. 



16 (?. Minrichs on Planetology. 

come uniform in the successive homothetic ellipsoidal shells in- 
cluded between any two successive surfaces. Hence we may 
consider ^ as a function of the equatorial axis of such surface, 
and, as the density is increasing toward the center, we may, in- 
stead of the general law 

S=f(a), .... (16) 

take the law assumed by Laplace for the interior of our earth, 

^ = A-ca. . . . (17) 

Plana has shown' that this law is the most probable in the 
case of the earth. Prof Forchhammer of Copenhagen has lately 
shown^ how this law accounts for one of the principal circum- 
stances relating to the succession of geological strata. 

It must finally be borne in mind that the nebula may have the 
various elements at the same place, because the laws of diffusion 
of gases will appl}^ to the gaseous nebula. Thus far the chemi- 
cal analyses of meteorites and the spectral analysis of the sun, 
moon and planets have corroborated this conclusion; still we 
must not hastily conclude that some differences mi ay not obtain, 

* Note sur la densite moyenne de I'ecorce superficielle de la terre. Asironomische 
Nachrichten, 1851, vol. xxv, No. 828. For the density at the center of the earth, 
he finds 16'3010. 

^ Inledning til en Rsekke af Forelpesninger Stoffernes Kredslob i Naturen (on the 
circulation of the elements in nature.) Nordisk JJniversUets-Tidskrift, viii, 1 
hefte. Copenhagen, 1862. p. 68-81. 

As an instance, Forchhammer desciibes the circulation of Lime: first the smaller 
diurnal orbits betVeen plants and animals and the soil; next the greater annual cir- 
culation between land (washed by rain) and the sea; and, being here deposited 
in large strata through the agency of marine animals, these are again, after long ge- 
ological periods, put into circulation, especially by the inorganic powers of nature, 
for these again to recommence a new cycle,— Now, where did the lime at first come 
from? Forchhammer thinks that as granitic rocks (specific gravity 2"*7) ar€ less 
dense than the dark trap-rocks, (on Bornholm, sp. gr, up to 2-93,) they would, during 
the first period of the igneous earth, float upon the trap; thus the first solid shell 
would be formed of granitic rocks, i. e. free from lime — consequently unfit to sup- 
port life, not fossiliferous. By the next revolutions this rock was dislocated and 
broken through by the underlying trap-rocks containing lime and iron as silicates; 
the atmosphere being so rich in carbonic acid and being dissolved in the then hot 
waters would decompose these silicates, and thus bring lime into circulation. Or- 
ganic life can now first commence — and the first fossiliferous rocks" appear. This 
beautiful idea is further substantiated by the fact that volcanos, after a longer 
period of rest, commence their eruption with emitting trachytic, i, e. granite-like, 
masses almost free from lime — which are later succeeded by the heavy black lava, 
containing both lime and iron. 

The richest deposit of gypsum, the Triassic period, is succeeded by the extraor- 
-dinary limestone formation of the Jiirasslc period, thus giving another link in this 
'Chain of inductions, for gypsum, beis^g more soluble, will more rapidly circulate, 
and thus occasion a greater deposit of lim^estone. 

These views of our illustrious teacher show us what patient investigation yet may 
accomplish; we see the cause for the succession of granite' and trap — see why or- 
ganic life could not commence earlier than it does — see the cause for abundance of 
limestone during tlie Jurassic period, etc. — in the simple circumstance that granitic 
'^masses, being lighter than trap, were exterior to the latter in the igneous globe. 



G. Hinrichs on Planetology. 17 

as the diffusion certainly is limited by the sinking of the denser 
particles. In a nebula from which a whole cluster of solar sys- 
tems has been formed, we may therefore expect to find consider- 
ably different elements. We thus decline the imputation of 
Rutherfurd that homogeneity of original diffuse matter " is almost 
a logical necessity of the nebular hypothesis," and cannot see 
any real objection to this hypothesis, if, as he says, " we have now 
the strongest evidence that they (the stars) also differ in constit- 
uent materials" (this Journal, 1863, vol. xxxv, p. 77). 

In regard to the signification of ^ we must remark that, in the 
following, we use the letter 8 to represent the mean density of the 
nebula from the centre to the distance r, while in (17) ^ indicates 
the density of* the shell at the very distance r. As (17) is only 
adduced to serve for a comparison, this course is legitimate. 
But it is easily demonstrated, that, at least for a spherical nebula, 
this law (17), if true for the individual shell, will also be true for 
the mean densitj^ of all shells inside of it. For, the actual den- 
sity varying according to (17), the mean density of the interior 
body from ?-=0 to r is found to be 

d — J-c'.r, (18) 

where c'=Jc. This law* is evidently the same as (17). 

§ 9. Attraction in the Nebula. 

As the nebula now may be considered made up of homothetic 
oblate ellipsoidal shells, individually of constant density, and as 
we know (from Mec. Cel., liv. iii, chap. I, § 2,) that such a shell 
does not exert any attraction on a point within, we find the attrac- 
tion at any point in the nebula determined by the attraction of the 
ellipsoid whose surface passes through that point. 

This force, at the point a?, y, z, is given by the following for- 
mulae (from Mec. Cel., liv. iii, ch. I, §4), independentof the law 
of the density (17), and merely depending on the proved uniform- 
ity of the densitj'' in each separate shell. 

If we put 

2i3['--'^-i-:prJ • • • • • (19) 

then the components X, Y, Z, of the attraction (positive toward 
the origin) are 

X = Qf^; Y = Q.?1; . . . ,(20) 

X3 



Q: '" 



H<^-TiTH- (^^^ 



* The '* Density" in Trowbridge's article (this Journal, xxxviii, 354, 1864), is dif- 
ferent, because referring to the density at different periods of time. 



18 G. Hinrichs on Planetology. 

All of these three components act io condense "^Aiq nebula: but 
X and Y also determine the revolution of the particles, while Z 
has no such influence, all motions in the direction of the axis of 
z mutually destroying: each other, because a-, y is the invariable 

plane. Composing X and Y we get 

R = Q.-3' • (22) 

and directed toicard the axis of rotation ; r^ —x- +?/-. 
Substituting the first {!?>) {rn = M)'in (22) we obtain 

R-f-iJr, 
vfhQve 



f=^^[(.^+^')^'-^ (<?='■)-'■]■ 



(23) 



As now « on]}' depends on /-, i. e. on the eccentricity (10) which 
is constant, the shells being homothetic, we see that « is at any 
given moment for all parts of the nebula the same, hence: the 
radial force R in the nebula is propO}iio?ial to the density 8 and the 
distance r from tJie axis of rotation. 

This simple result is of very great importance, as we shall see 
in the sequel 

§ 10. The orbit of the Planets. 

The particles of the nebula had originally motions in all di- 
rections; but as we assumed the existence of a momentum of 
rotation (§ 6), the principle of the invariable plane will keep up 
this momentum (§8), while all motions at variance therewith will 
in time mutually destroy themselves. Therefore, all p)articles 
describe circles around the axis of roto.tion. 

Such as the orbit of the single particles that formed a planet 
will also be the orbit of the latter : hence the eccentricity and 
inclination of all planetary orbits ought to be zero. This may 
also be seen from Plateau's Experiment, and agrees well with 
the smallness of both the eccentricitv and the inclination (see 
§1, 9' and 11^ § 3, 4° and 5^). Stilf, neither of these two quan- 
tities is actually zero. Are, then, these small deviations from 
this value accounted for by some accessory conditions of the 
problem ? 

We think so, for there are tico modifying circumstances, the 
rupture of the ring — which it is beyond our power as yet to take 
into consideration — and the perturbating influence of already 
separated masses. The latter we may estimate. Representing 
the eccentricity by e, the inclination of the orbit to the ecliptic 
by ?', to the invariable plane by I, we have from observation 
[HumboldCs Cosmos'] :^ 

* The numbers in the last column of the following table are not quite exact. — 
Ei>s. JouE. Sci. 



e. 




%. 




I. 


■2056 


7° 


C 


6° 


19' 


•0068 


3" 


23' 


1° 


42' 


•0168 


0" 


0' 


r 


41' 


•0932 


1° 


51' 




10' 


•160 


7° 


55' 


6° 


14' 


•0482 


1° 


19' 




13' 


•0561 


2° 


30' 




48' 


•0466 


0° 


46' 




55' 


■0087 


1° 


47'. 




6' 


.... 


1° 


41' 


0= 


• 0' 



G. Hinrichs on Planetclogy. 19 



Mercury, - - - 

Venus, - - - 

Earth, 

Mars, - . . 

Asteroids,* - - - 

Jupiter, - - - 

Saturn, - - - 

Uranus, - - - 

Neptune, - - - 

Invar, plane, 

We see how clearly the principal members of the system 
move in one 'plane ^ and that this plane is the invariable plane of 
the system ; the great planets deviate less than one degree, the 
principal of the interior planets, Earth and Venus, only If de- 
grees — and even the inclination of the smallest planet, Mercury, 
amounts to but 5-J degrees! So also in relation to tbe eccen- 
tricity, this being less than one-twentieth for the principal bodies. 
As to the deviations, we see that Keptune, which if not the 
most distant planet, certainly is (or was) separated from the next 
by a very large distance, so that if either could not at all, or but 
slightly, be disturbed, has indeed the smallest inclination (only 
6 minutes!) and about ihe smallest eccentricity (less than one- 
hundredth !). Jupiter, which, on account of its enormous mass, 
could not be much disturbed by other bodies, has an inclination 
of only 13 minutes, while Saturn and Uranus have — correspond- 
ing to their smaller mass — about four times as considerable an 
inclination (48 and 55 minutes). The eccentricities of these 
three orbits are about equal ; perhaps that of Jupiter is near its 
maximum, or the eccentricity of Saturn and Uranus near their 
minimum. 

The inclination of the Earth and Venus is greater than that of 
the exterior planets, for the mass of the former is small as com- 
pared to that of the latter ; but as Venus and the Earth are the 
great planets among the interior, we see that the inclination and 
eccentricity of Mercury's orbit are much more considerable than 
either, and that Mars has less inclination and eccentricity than 
Mercury. Is it because Jupiter, the only planet that would exert 
considerable perturbation on its development, was so far distant? 
The orbit of the asteroids is explained in § 5, V.^ We gave 
publicity to these views in an address delivered before the phys- 
ical section, at the meeting of the Scandinavian philosophers, 
July, 1860. 

' Mean of the first 72 Asteroids, elements given in Table of Smithsonian Report, 
1861, p. 218-219. 

' We intended in this place to give a fuller account of our views concerning the 
development of the asteroids ; but learning from a letter of Mr. Trowbridge that 
the continuation of his article will contain a solution of this problem, I abstain for 
the present from publishing my details. 



20 G. Hinrichs on Planetology. 

The same principles will apply to the satellites ; but we have 
too few data to make a comparison of this principle with obser- 
vation profitable. 

§ 11. The periodic time of ike Planets ; Kepler's third law. 

Since every particle in the same shell revolves around the axis 
under the influence of a force K proportional to the distance r 
from the axis (§ 9), we know from mechanics that the periodic 
time T of such ^ particle is 

^=;& <"' 

where, it will be remembered (23), /* is the same for the whole 
nebula, and <^ constant for the same shell, so that the time of rev- 
olution is the same for all particles of the same homothetic shelly hut 
for the different shells inversely proportional to the square root of the 
density. Thus every shell rotates as if it were solid ; and if the 
whole nebula had the same density throughout it would rotate 
like one solid. But if the density be different in different parts, 
some shells will rotate faster than others (§ 12). 
Eliminating S by means of (13) we get 

«3 ^ 



M 

8 A3 



(25) 



3 \/l+A2 (i_[_^2)tan-U-;.^ 

We know that the ellipticity of the nebula is determined by 
the centrifugal force, and the latter by the state of condensa- 
tion (§ 8) ; and even in case an ellipsoid becomes impossible, we 
can not but conclude that the figure continues to be determined 
in the same manner. But the condensation continues — the in- 
crease of the centrifugal force depending thereon will also con- 
tinue and produce a series of rings in a certain succession, just 
as one ring was formed in the experiments of Plateau (§ 7). We 
see now how the continued increase of the condensation occa- 
sions a periodical change in the figure of the nebula. Grranting 
the variation of the figure beyond the possible ellipsoid to be 
determined by the same circumstances as the ellipsoid itself, we 
may compare the corresponding stages of the nebula by referring 
to the same ellipticity e or the same ,<^' in (25) ; at any rate, we 
know that this can.be done if we only compare the nebulae, when 
within the limits of the possible ellipsoid. But then .«' will be 
the same for all rings, and as the mass of the planets is but very 
small as compared to that of the sun, M remains almost constant. 
Then (25) becomes 

--= constant; (26) 



G, Hinrichs on P lane t o log i/. 21 

or the squares of the times of rotation of the different rings are as the 
cubes of their radii. 

If we remember that the possible ellipsoids reach to a propor- 
tion of 1 to about 3 between polar and equatorial diameters of 
the nebula, we can be sure that tliis covers the principal part of 
the metamorphosis ; hence, (26) is rigorously proved for the 
greatest part of the condensation intervening between the fornm- 
tion of two successive rings; the nebula acquires its pi'iiicipal 
dimensions while changing in accordance with the ellipsoidic 
figure, and when abandoning this it quickly passes to the form 
of a slightly oblate spheroid and a ring. The interruption in 
our strictly mathematical demonstration, cannot, therefore, seri- 
ously interfere with (26). But then tiiis or Ktpler^s third laio is 
a consequence of the nebular h(/pot'ie-'ii:^^ or t/i.e ob,:<ervations en'd>odied 
in this law sustain equally the nebular hi/polhesis and grar^itation. 

Again, inductivel}', we may conclude from Kepler's third law 
that the interruption in our analytical deductions occasioned by 
our ignorance of the exact mechanical laws of the metamorpho- 
sis of the ellipsoid into the r/lobe ring (we might in reference to 
Saturn find the expression Kronionform convenient) is not of 
serious consequences. 

Thus we may at least conclude from the third of Kepler's 
great laws thai the development of the planets was periodical ; 
for, this law being a fact, and (25) being rigorously true, we must 
have 

-- = constant; (27) 

but, as remarked, before, M remains essentially constant, hence 
« or what is the same A, i. e. the ellipticity e of the nebula cor- 
responding to the different planets, must have been the same at 
corresponding epochs, just as we assumed above. 

But // the metamorphosis of the nebula has been periodic, and 
not simultaneous, we must ascertain wliether die successive inter- 
vals of time were equal or not. We shall find that dicy loere equal^ 
just as it would be the most natural or the simplest to assume. 

§ 12. Spiral jVebidce. 

In the preceding paragraph we considered the density of the 
nebula sensibly equal throughout, so that the nebula always ro- 
tated like a solid, all particles having sensibly the same period 
of revolution. This might be done, because \he dimensions of 
such nebula — however immense in reality — are not sufficiently 
great to produce a very large change in d (17) in the space al- 
lotted to each planet. 

But there may be bodies of dimensions so vast as to render it 
utterly impossible to consider the density approximatively uni. 
Am. Jour. Sci.— Si5coxt> Seriks, Vol. XXXIX, No. 116.— March, 1865. 
9 



22 G. Hinrichs on Planetology. 

form throughout the nebulous mass. Then the nebula will not 
rotate like a solid, but the angular velocity w of any particle 

will be ^""t"' (^^) 

or, by (24), a) = Vjd. (29) 

As fi (23) is constant for the whole nebula, we see that the an- 
gular velocity is proportional to the square root of the density, or, 
according to (17), greatest near the center of the nebula. 

If 6 be the angle of position of those particles which are 
originally (i. e. when t=0) in one and the same straight line, we 
have at the time /, 

dzzzojt^ (30) 

or by (29) 

d2=zfi.d.i^, (31) 

Kemembering,that the density is a function of the distance 
(16) and also of the time on account of the progressing conden- 
sation, we see that (31) may be written, 

02 =\ut2f{a, t) (32) 

At any given moment of time {t constant), all the particles 
that originally were situated in the same straight line given by 

^=0 (33) 

will now form the curve 

62 = g> (a), 

i. e. a spiral. This contains the fundamental principles of a me- 
chanical theory of the spiral nebidcB. 

Substituting Laplace's law of the density (17) in (31) or (32)^ 
we obtain as the equation of the spiral 

«=-— C— ' (34) 

wherein az=ju.A.t- depends upon the ellipticity (^/), the density 
A at the center, and the time i, whilst C =cf^t" depends, upon the 
same f^ and t and the rate of variation of the density. We see that 
these spires are limited, for 0<^<«; and that the sweep « of 
the spire increases with the age of the nebula, the density at its 
center, and the ellipticity. 

In order that such spiral structure may become apparent in a 
regular ellipsoidal nebula, the brightness must originally have 
been different in different meridians, though the density was 
constant in the same shell, i. e. the same in all meridians. Thus, 
if the brightness in the nebula was originally greatest in the op- 
posite meridians AC and BC, (Fig. 1.) and it rotates in the di- 
rection of the arrow around the axis C, the spi^^al iiehida, (fig. 
2,) would result. As the age increases, the sweep, or the angle 



G, Hinrichs on Planetology. 



28 



BCE = «, would increase, whilst A and B remain nearly at the 
same distance from C : so that an annular nebula with a central 
core might in time result from a spiral nebula; even several 
concentric rings might be formed. 

1. 2. 





We cannot suppose any nebula to have different brightness in 
parts of the same density ; and neither is it reasonable to assume 
such vnst masses to be already shaped to a regular ellipsoid by 
the influence of the central forces (see 7). 

It is much more reasonable to think that the nebulous mnsses 
at first were of any shape — such as might result from a predom- 
inatinoj attraction of tliose portions where the heaviest elements 
vvei'e formed or collected in greater abundance. Then the for- 
mulie deduced in the preceding paragraphs, though no longer 
representing the exact conditions of the nebula, still would con- 
tinue to be approximate; the angular velocity would still be 
greatest near the central parts, as can also easily be shown di- 
rectly, by considering the motion of each particle as subject to 
the attractions of all the others. Then the particles originally in 
a straight line would still in time form a spiral. 

So we see that a nebula originally in the shnpe of a light rec- 
tilinear cloud with a condensation near the middle, like the part 
AB in fig. 1, would after some time exhibit a spiral like the dark 
part in fig. 2. The nebulae, Herschel 1061, and H. 1337, as seen 
by Lord Rosse,' have exactly such a form. If, instead of having 
the nucleus in the middle, the original nebula had been denser 
near one extremity, like fig. 3, a simple spire like fig. 4 would 





be the resulting spiral nebula, as we see it in H. 327, H. 1946, 

^ Prof. G. P. Bond, Director of Harvard College Observatory, kindly sent me 
copies of a number of Rosse's latest figures of spiral nebulae— for which important 
service I here repeat my sincere thanks. 



24 G. Hinrichs on Ptanelology. 

etc. A nucleus with four branches of different density and 
magnitude would give a spiral nebula like the beautiful object, 
Messier 99. If each of the two arms in figure 1 had been sub- 
divided into two branches, H. 2084 would result. 

These few remarks must be sufficient at this place. TVe have 
already deduced forms as fanciful as H. 1196, H. 131, H. 1744, 
and others, from simple rectilinear forms, and we hope before 
long to discuss this theory more at length. We here merely in- 
tended to show that the forms revealed to us by the great tele- 
scope of Lord Rosse appear to be simple mechanical consequen- 
ces of the nebular theorj^, if applied to very large nebulae. 

§ 13. The Law of the Planetary Distances. 

The law of the planetary distances has not as yet been discov- 
ered, though it has been most diligently sought for as the prin- 
cipal element of the " Harmon}^ of the Spheres." The endeavors 
of Plato were in vain, and Kepler at last ascended to the truth 
that the present distances are not exactly the original ones. 
Titius, and after him Bode, came near it [§ 2, (1)] ; but the de- 
viations from this law remained unaccounted for, thus not giving 
the conformation most essential to any law. 

To find the true law^ of the planetary distances has been our 
aim for nearlj^ ten years; we hope the sequel will prove that we 
at length have found the solution of this problem in the follow- 
ing law : 

Tke mutual distances of the planets correspond to equal intervals 
of time. 

That this is ^fact we will demonstrate; but v:hy these inter- 
vals were equal we are not yet able fully to see — still we know 
that this is the simplest way in wdiich the periodicity in the de- 
velopment of the nebula as found in § 10 can obtain. 

Deferring a thorough discussion of the earlier attempts, (some 
of which are almost contemporaneous with our own solution,) to 
some future opportunity, we will now give the inductive reason- 
ing which leads to our law above stated. 

There are a few well known laws in the evolution of the neb- 
ula which em.bodv the solution of the problem. We know that 
th:" planetary masses are insignificant as compared with the solar 
mass ; hence we see that the orbits of the planets simply mark 
the equatorial band of tlie condensing nebula at those definite 
periods when the radius of the nebula had diminished to the 
distance of the planet. Thus we see that the plo/netary distances 
mils:/, he functions of time. 

Or, if it be more plain, we may say that the original nebula, 
in contracting, left at certain intervals a few particles behind to 
mark the limit of the nebula at those instants. But while con- 
densing, the uttermost panicle of the nebula describes a spiral 
.curve: and if we can find the relation between the distance a of 



G. Ilinrichs on Planetology. 25 

this particle and the time /, we need only to substitute the different 
intervals corresponding to the formation of the different planets 
in order to obtain tlieir distances. But as ihe evolution is regu- 
larly periodical (§ 10), it is most probable that these intervals are 
equal; comparison with observation shows this to be the case. 

But the motion of such a particle in so rare a nebula is regu- 
lated by the attraction of the whole nebula and the resistance of 
the ether. The first of these forces is inversely proportional to 
the square of the distance a, since the mass remains sensibly 
the same, and the particle is considered as on the equatorial sur- 
face of the nebula. In other words,, the force of attraction on the 
particle is the same as the force of gravitation acting upon a 
planet. Kesistance of the ether will necessarily follow the same 
law, whether a single particle or a planet be subject to it. But 
then our analysis^ of the motion of a planet toward the sun is 
directly applicable to the motion of the superficial particle in its 
fall toward the center of the nebula. Formula (10) of tliat ar- 
ticle shows the distance a (radius of the nebula) to be 

a—K.s'"^"^ (35) 

where A is the original distance (or radius of the nebula) and t 
the time of falling from A to a. Or, if at represent the distance 
of the planet that separated from the principal nebula at a time t 
earlier than the now nearest planet — i. e. the age of the planet as 
counted from Mercury, — the above (35) becomes 

^cit = §.f (36) 

where § and y are constants. But in the analysis leading to (^b) 
the coefficient v of resistance 

3 5 

^-8^ (3') 

has been considered constant ; here we cannot do so, for though 
the density 8 of the ether and the radius ? of the particle may 
be considered constant, the density ^ of the particle varies very- 
much, about inversely as the cube of the radius of the (homo- 
geneous) nebula. Ifj therefore, v be the value of v correspond- 
ing to the particle at the distance 80-0 of Neptune, v" the same 
at the distance 04 of Mercurj^, we have for a homogeneous 
nebula 

v' : v'l =r (-4)3 : (30-0)3 _ i . 42200O 

nearly. If 5 increases toward the center this proportion would 
be diminished; but still we see that v decreases toward the inte- 
rior. The formula (36) can therefore only express the principal 
part of the law ; how (36) has to be amended in order to take 

' On the Density, Rotation and relative Age of the Planets. This Journal, 1 864, 

[2], xxsvii, 86. 



26 



G. Hinriclis on Planetology. 



the variation of t> into account has to be separately investigated. 
Before we attempt tin's we will compare (36) with observation. 

Erecting at equal distances (§10) ordinates proportional to the 
actual planetary distances, and interpolating by connecting these 
points by a curve, we see that this curve has the appearance of 
a logarithmic curve (like those given on the plate appended to 
our former article, this Journal, 1864, vol. xxxvii) ; and if drawn 
with sufficient care, we find that the constancy of tlie subnormal, 
characteristic of the logaritiimic curve, holds good in the present 
instance — thus proving that (36) really is applicable to the plan- 
etary distances. But it is not the exact law, for the axis in the 
diagram is evidently too far below the curve, or the distances 
are too great by a constant «, so that the diagram of the planet- 
ary distances will be expressed not by (36) but by 

atz=za-\~8.f, (38) 

and it remains to be seen whether this additional constant o. can 
be accounted for by the variation of ^^ (37). Before we investi- 
gate this, we will see how far (38) represents observation. 

We see that it is almost the same as the law of Titius, but 
while in the latter t is a mere index, it is in (38) a variable, the 
great independent variable of mechanics, time or age/ Besides, 
(38) deviates from Titius in the case of Mercury. Adapting the 
constants of Bode to (38), it becomes 

ai=r4-|-(l-5).2^ (39) 

Eepresentnig by a the actual distance, we have, for compari- 
son with observation, 

Distance 



Planet. 


age, t 


calc. Qf. 


obs. a. 


Difference. 


Mercury, 





55 


38-7 


+ 16-3 


Venus, 


- 1 


70 


92-3 


~ 2-3 


Earth, - 


2 


100 


100-0 


0-0 


Mars, 


- 3 


160 


152-4 


+ 76 


j^ steroids 0-@, - 


4 


280 


262-3' 


+ 17-7 


Jupiter, 


- 5 


520 


520-3 


- -3 


-Saturn, 


6 


1000 


953-9 


-1- 461 


Uranus, 


- 7 


1960 


1918-2 


-{- 41-8 


Neptune, 


8 


3880 


3003-6 


+ 876-4 



We see that the present distances a agree with the original Of for 
the principal planet of both groups, for the Earth and Jupiter. 
Mars, Sattirn, and Uranus are about ^-^th of their distance too 
near the sun, having approached the latter so much more on ac- 
count of their mass being smaller. Mercury and Neptune have 
even approached still more, the former because of the smallness 

" Calculated from the table in Smifkaonifm Report for 1861, p. 218-219. We 
founrl the following interesting fact: mean tiistance of (I) to (31) = 2 599 ; of (81) 
to (56) = 2-679 ; of (57) to (72) = 2 752, showing that in general the more distant 
membcra of the group of asteroids have been later discovered. 



G. Hinrichs on Planetology. 27 

of its mass, the latter on account of its high age (see this Jour- 
nal, vol. xxxvii, p. 41). Before the precise influence of resist- 
ance was known, tliese deviations were considered sufficient cause 
to reject the law of Titius-Bode; but now these very deviations 
have become essential supports of the truth of that law. 

Another and better test of our law (38), and of the constants 
of Bode (39), is obtained by directly solving (39) for the age t 

and seeing how far t is given by the series 0, 1, 2 . . . We thus find 

Planet. Age. too small. 

Mercury, - - - - imag. 

Venus, 1-1066 —'1066 

Earth, .... 2-0000 -0000 

Mars, 2-9056 +-094 

Asteroids 0-@, - - - 3-890 -\-'\\ 

Jupiter, 5-0001 -000 

Saturn, - . - . 5-9264 +-073 

Uranus, 6-9683 \--0^\ 

Neptune, , . - . Y'6230 -{■■^11 

From this table we see that the age of the planets abOve that 
of Mercury is as the series of natural numbers, the deviations not 
only being but sinall, bat just such as influence of the mass would 
make them. This may be easily proved by the formula con- 
tained in the article on the age of the planets before referred to. 

If the present age of Mercury be m, then the age of the inte- 
rior planets will be to that of the exterior ones as m + - is to 

26 ^ 

m + — , or as 2m. + H to 27?z + 13. We found this ratio as 1 to 
4 

3 (this Journal, vol. xxxvii, p. 43) ; if true it would follow that 

m z=z 1, or the total age of any planet ivould be t + 1^ the unit being 

the age of Mercury. 

After having seen that (38), the modified form of (36), is ap- 
plicable to the planetary distances, we will demonstrate that this 
modification is consistent with the signification of ^, the time. 

If the resistance R be proportional to the velocity v, or 

^=vv (41) 

we have the tangential force (this Journal, vol. xxxvii, p. 40) 
.dd\ 

=r — Rcos7; = -. JT— , . . (42) 

where r is the radius vector, and the anomaly; but Kepler's 
second law gives 

dd 




28 G. Hinrichs on Planetology. 

so that (48) becomes 

1 r^c , 1 

-r[dt + '^'\='^ ' (^4) 



giving/o?' V constant^ 



cmC.r"^, . (45) 



or, since bj Kepler's third h^w, c^=a^^ 

a^2!.-2^^ . (46) 

all of which formulae are demonstrated in our article referred to 
above. 

Instead of solving the problem directly, we rnay indirectly try 
to find how v must vary that (36) may become (38;, i. e. to add 
a constant term to (46). In other words, C instead of being con- 
stant must be considered a function of ^, i. e. (44) must be 



1 Tde , 1 



so that the resistance now becomes, see (42), 

R=zvv--^-^=zvv'- --.^-^, .... (48) 
cosy do r 

instead of (41), where cos ?; = -—, and ds is the element of the 

orbit. The function cp{t) can now, by the method of the varia- 
tion of the arbitrary constants, be so determined that (46) or (36) 
coincides with (38). Since r is a function of t, we may make 

f{t) = rcp(t), (49) 

hence (47) becomes 

S+-=/w (^°) 

Taking the complete differential of (45), i. e. also considering 
C variable, substituting in (50) and reducing by (45), we obtain 
for the determination of C, 

-vi dO 



e 



dt 



■■m (51) 

This gives, by making K an arbitrary constant, 

Qz:zK+fe'Kf(t)M, (52) 

which, substituted in (46), gives, 

a = !l!l'[K+/.'"./W.<?<]=. . . (53) 
r 

This should be identical with (38), i. e. (remembering that t 
here is counted from the most distant, in (38) from the nearest 

planet and that y in (36) is e " in (35) ) 

a=ia-\-§,e~ (54) 



G. Hinrichs on Planetology. 29 

Equating (53) and (51), and solving ioTf{t), we find, 

a v/IZ 
f(t) = v^-= .... (55) 

or, by (54), f{t) = vuV (56) 

But Kepler's third law gives 1.1^=1 a. v^ (this Journal, vol. 
xxxvii, p. 38, note); hence 

f{t)=^av, (57) 

consequently, by (49), r and a being now the same again, 

^{t) = va'L^ (58) 

or (48), cos V being almost equal to one, the orbit being nearly 
circular, Rz^i^i'jl J (59) 

Thus we see that (36) becomes (38) if the resistance E, instead 
of beino- simpl}^ proportional to the velocity (41), is varying ac- 
cording to (59), which may be comprehended in (41) by taking 
the factor v to decrease from v{a= -j^) to (a = «) according to 



-'('-"a) (^°) 



This variation of the coefficient of resistance is conformable to 
(37), since ^, according to (16) (then (J), increases as a decreases. 

The law at = a^^.e =a-\.§.y' 

is, therefore, but an amplification of 



at = §.e 



■2n 



in the latter the coefiicient of resistance is constant^ in the former 
it varies according to (60). As now (60) is real, (54) or what is 
the same (38) is the real law of the planetary distances, t contin- 
uing to represent the age, and not, as in Bode's law, a mere in- 
dex. And as now finally (38), applied to the actual distances^ gives 
values for t that are very nearly as the natural numbers^ our law, 
announced above^ holds true^ that the planetary distances correspond 
to equal intervals of time ; or the consecutive planets vjere abandoned 
at equal intervals of time. 

There remain yet two remarkable consequences to be drawn 
from this exponential law of the planetary distances. If in (38) 
it is sufficiently great (i. e, the corresponding planet far from the 
center) to make the first term insignificant as compared to the 
second, we have approximatively 

a =8.f, 
hence a^+ii= (9.7f+i, 

Am. Jour. Sol— Second Series, Vol. XXXIX, No. 116. —March, 1865. 
20 



30 G. Hinrichs on Planetology. 



consequently J±l::^y^ /gl\ 



or/or the most distant planets their distances approach to a simple 
geometrical series whose quotient is the base /. But this law will 
again in part be interfered with on account of the action of re- 
sistance on the completed system, which, on account of the 
higher age, lias drawn the most distant planets comparatively 
nearer to the sun than the less distant ones, so as to diminish the 
above quotient /. 

Again, if t is sufficiently small — or the planet sufficiently near 
the center — the exponential series contained in (38) is highly 
convergent, so that perhaps the approximation may be sufficient 
if only the term of the first order is taken, so that (38) becomes, 
A and B representing constants, 

ff; = A4-B.^, (62) 

hence a^+i = A+B(i!4-1), 

r/,+2 = A+B.(^-|-2), etc. 
or, c/,+ , -a^rz:a,+2-«^+j=:B, . . (63) 

i. e. the innermost planets have a tendency to become equidistant. 

Both of these consequences are very plainly marked in the 
solar system, especially in the lunar, but also in the planetary 
orbs. For, as regards (61), we have for the distances of 

Saturn to Jupiter as 1-85 to 1. 
Uranus '' Saturn " 2-01 " 1. 

Neptune " Uranus " 1-57 " 1. 

For Saturn-Jupiter this proportion is still less than y = 2 — also 
because Jupiter both on account of its age and mass has fallen 
less toward the sun than Saturn ; but for Uranus-Saturn the 
ratio is almost equal to y = 2, while for Neptune-Uranus it is 
less again, on account of the higher age of the first. 

The second circumstance, expressed in (63), seems to be exem- 
plified in the orbits of Mercury, Yenus, Earth, the three planets 
that are nearest to the sun, or for which t is the smallest. Their 
distances are 

Distance. Difierence. 

Mars, .... 152-4 

Earth, .... lOO-O ^^ ^ 

Venus, .... '72-3 ^^.^ 

Mercury, - - - 38-7 ^^ ^ 

We see how Mars, Earth and Yenus follow Bode's law exactly, 
for one-half of 62'4 is 26*2, or very nearly 26*7 — but the distance 
between Yenus and Mercury is 33'6 instead of J of 26-7 or 134. 
This difference miight be considered as a consequence of (63) ; 
but we know that it is principally due to the small mass of 
Mercury. 



G. Hinriclis on Planetohgy. 31 



§ 14. The Lunar Distances. 

As Keplers third law was deduced from the planetary orbits 
alone, so was the law of Titias. But it was shown to be a con- 
sequence of the law of universal gravitation, and therefore itself 
universal and applicable to any system — hence, also to the lunar 
systems, Now the law of Titius, as modified above, has been 
found to be identical with the equality of the intervals of time in 
the history of any system. Therefore, also, this law (38) must 
apply to the lunar systems. This we now will show. 

A. The Lunar System of Jupiter. 

The Jovial World is the youngest of those great lunar systems 
that adorn the exterior planets. (This Journal, xxxvii, p. 45.) 
Therefore, it is the most regular yet of any, and our law (38) 
must very closely harmonize with the actual distances of Jupi- 
ter's moons. It is easily found that /=2, again, as for the 
planetary distances; and that "=4 and |9=3 radii of Jupiter. 
Thus (38) is for the Jovial World, 



Moon 





^i 




stance. 






t. 


Calculated. 


Observed. 


Fall. 


T. 





V 


6-049 


•951 


II. 


1 


10 


9-623 


•377 


III. 


2 


16 


J5-3o0 


•650 


IV. 


3 


28 


26-998 


1-002 



(64) 



The ''fall" of a moon is the distance it has fallen toward the 
planet in virtue of the resisting ether. That this fall corresponds 
to the age, mass and density of the different moons has been 
shown in our previous article. (This Journal, xxxvii, 45.) 

The calculation of t from the observed distances gives for the 
2d, 3d and 4th, respectively, -907, 1-92, and 2'94, which only 
deviate by "09, "08 and -06 from the theoretical values 1, 2 and 
3 ; and all values being too small shows that these moons are 
correspondingly nearer the primary, having approached so much 
on account of the etherial resistance. 

B. The lunar world of Saturn 

is next in age, hence not quite so regular as that of Jupiter. 
We find that (38) represents the distances of the eight moons if 
the constants are 

a^rz:4+0•35X2^ (65) 

as will be seen from the following table : 



32 G. Hinrichs on Planeiolo^i 





Distance 


. 




Moon. 


Calculated. 


Observed. 


Difference. 


I. Mimas, 


. 4-3 5 


3-4 


+ •95 


II. Encejadiis, . 


4-70 


4-3 


-J--40 


III. Tethvs, 


. 5-4 


5-4 


•0 


lY. Dione, 


6-8 


6-8 


•0 


v. Ehea, . 


. 9-6 


9-6 


•0 


VI. Titan, 


. 15-2 


22.2 


-7-0 


Vn. HyperioQ, 


. 26-4 


28-0? 


-1-6? 


^III. Japetus, . 


. 48-S 


64-0 


-lo-2 



Excepting for a moment the 6tli and 8th moon, we see but 
small differences; Mimas and Enceladus being too near Saturn, 
appear to have but very small mass, which, conclusion is strength- 
ened by the fact that it required Herschel's great telescope to 
discover them (1789). The next three almost exactly harmonize 
with this law ; they are, therefore, not only larger than the first 
two, but also much alike. Thev were discovered by Cassini, 
first the fifth (Ehea) in 1672, 'and later (1684) Tethys and 
Dione. As the latter were discovered by the same observer, the 
difference in date is, perhaps, alone due to the greater nearness 
to the disk of the primary. Hyperion is even lower than any, 
and, therefore, smaller than even the interior ones. This is con- 
firmed by its discovery, which was not made till 1848, by Bond 
and Lassell. But the sixth. Titan, and the eighth, Japetus, are 
much farther distant than {Qb) gives; thus proving them to have 
much more considerable mass (or rather v (37) is less, which in 
general will be the case if the mass is greater). This is fully 
confirmed by the date of discovery : Titan being the first dis- 
covered of all, (by Hu3^ghens, 1655), and Japetus the second, (by 
Cassini, 1671). These estimates of the masses are further cor- 
roborated by Humboldt,* who calls Titan " the largest of all 
known secondary planets." Compare another theoretical esti- 
mate, (this Journal, xxxvii, 46), leading to the same results. 

C. The lunar system of Uranus 

is exceedingly important on account of the plane and direction 
of its motions. We have tried to show that this very position 
affords one of the most conclusive confirmations of the nebular 
theory. (This Journal, xxxvii, 50.) Here we will consider 
the arrangement of the individual members of the system. 

TTe know it to be the oldest, because it is the most distant 
system of which we have definite knowledge. The original dis- 
tances and the original harmony of these distances is therefore 
here most deranged. We cannot even v/ith any degree of cer- 
taint}^ consider the moons to be now in the same order of suc- 

^ Cosmos,!, Harpers edit, p. 95. 



G, Hinrichs on Planetology. 33 

cession as at first. At tlie same time, observation lias as yet 
hardly determined tlie number, much less the exact distance of 
the different moons. Therefore, we give the following more for 
the sake of completeness than with the view of adding any im- 
portant confirmations of our law. 

We have seen that the nearest luminaries may be equi-dis- 
tant, and that the farthest may succeed at distances that form a 
geometrical progression — see (61) and (63). If the distances, as 
given by Herschel, and the times of revolution, as given by 
Lassell/ are exact, we may represent the distance of the first six 
moons by 

at—1-b+^t, {<dQ) 

corresponding to (62), and the distance of the sixth, seventh and 
eighth by 

at^a^.2t-\ (67) 

corresponding to (61). 

Distance. 
Calculated, 

Moon 



Calculated. 


Observed. 


Difference 


l—1-b+OX 3 =: 7-5 


7-5 


•0 


II=7-5-f-lX 3 =10-5 


10-5 


•0 


III=:-5-f2X 3 =13-5 


131 


+ -4 


IV=:Y-5-|-3X 3 zz:l6-5 


17-0 


- -5 


Vz=7-5-|-4X 3 —19-5 


19-8 


- -3 


VI=7-5-|-5X 3 =22-5 


22-7 


'2 


VII— 2X22-5r=45-0 


45-5 


- '5 


VIIIizz 4X22-5=:90-0 


91-0 


-1-0 



If these observed distances really are correct, then this re- 
markable discontinuity will enable us to determine the lunar 
masses long before observation can ascertain them. 

D. Conclusion. 

The lunar system of the Earth, consisting of but one moon 
and that of Neptune, which comprehends one or two, cannot, or 
do not afford any chance to test our law. But we have seen that 
the systems of Jupiter and Saturn fully confirm our law (38); if 
due regard is had to the individual mass and volume — or 
density and radius — of the several moons. Even the system of 
Uranus, as far as known, does not deviate from it except in so 
far as it offers the two extreme limits of the law, probably on 
account of the high age and a close similarity between the masses 
of the first six moons. 

Therefore we may say that as far as observation on the lunar 
systems goes it is embodied in our law (38), or m every lunar 
system the consecutive moons were formed at equal intervals of time. 

^ See Schweigger in Astronomische Nachrichten, No. 832, Beilage. 



34 G. Hinrichs on Planetology. 

§ 15. The incommensurahility of the 2^eriodic times. 

By the third law of Kepler we have, if T^ and T^^ are the 
periodic times of two planets, 

T=[^r^ (^^) 

or by (88), 

T-L.-f-;¥J' ^ ^ 

which expression will not generally make T^' and T^ commensur- 
able. Thus we see that our law accounts for another important 
condition of stability of the system, (see § 3, 1). 

But as the distances are continualh' decreasing, and at differ- 
ent rates, (this Journal, xxxyii, 41, gives the numerical values 
of these rates), we perceive that in time such commensurahility 
may take place between any two planets." Such is actually the 
case between Jupiter and Saturn, as discovered by Laplace. 

The distances were (see § 13) for Jupiter a. = 520, for Saturn 
<Zg = 1000, giving for (68) the continued fraction 2(1, 1, 2 . . ) 
having the approximations, 

2 3 5 13 

1' 1' 2' y • • • 

or T g : T . approached originally to 5 : 2 ; now it is ver}^ nearly so. 
For Yen us and the Earth the original distances 70 and 100 
give the approximations, 

1 2 5 12 29 

1' 1' 2' y TY • * • 

whilst Air}^ has found the commensurahility 13 : 8 or nearly our 
29 : 17 [13: 8=29 : 17-8]. 

In the lunai^ systems such commensurability is common ; and 
it is for the satellites of Jupiter that Laplace demonstrated "^ the 
great proposition, (f such commensurability exists hut approxima- 
tively it will become exact in time. 

Having seen that the change in distance produced by resist- 
ance will make the ratio approach commensurabilitj^, it therefore, 
as we stated before, will become rigorously so. • 

From (68) we find easily that the ratio will be 2 if the dis- 

3_ 

tances are in the ratio of 2^ : 1, or (by continued fractions) as 
the approximative fractions, 

1 2 3 8 19 27 



^ Grant {History of Physical Astronomy, London, 1852, p. 93,) states that the 
libration of the jovial moons is " independent of the effects of a resisting medium," 
meaning tliat it will be preserved notwithstanding such medium. This is probably 
a mistake, for it would depend upon the relative magnitude of the resistance and 
the perturbation. 

* Mec. Cel., vol. viii, Ch. vi, § 15. We express the proposition in more general terms. 



G. Hinrichs on Planetology. 35 



For Jupiter's satellites we have fl!2=16j ^1=10, or fl 2 : aj = 8:5; 
and «! : a^ = 10: 7=8: 5*6, therefore we find the periodic time 
of the second moon twice that of the first, Tj= 2To, and the 
periodic time of the third twice that of the second, ^^^=z2iT^ ; 
hence Laplace's famous relation between the mean motions, 

In the system of Saturn similar relations obtain. At first we 
had (see § 14, B) for the distance of Tethys (III) and Mimas (1) 
the ratio 64 : 44z=8 : 'o'Q^ while the duplication of the periodic 
times requires the ratio 8 : 5. But Mimas has approached Saturn 
the most, and thus this proportion (now 5'4 : 3*4=8 : 5*04) has 
been brought about. 

For the fourth and second we had originally, Dione : Encela- 
dus=6*8 : 4'7=:8 : 5"0, or likewise suificiently near 8 : 5 that the 
duplication of the periodic time should become almost rigorous.^ 

The lunar world of Uranus is particularly noted for such 
duplications, from the fact that Schweigger, as early as 1814, on 
such grounds predicted the existence and gave the orbits of the 
two innermost moons of Uranus, which were discovered by Las- 
sell in 1851. The coincidence is very remarkable, as will be 
seen from the following:^ 

Schweigger, 1814. Lassell, 1851. 
Uranus, I moon, - - - 2*1767 days, 2-5117 clays. 
II - - 4-3534 " 4-1445 " 

and the lY (or II of Herschel) having a period of 8'7068 days 
approximates to the further duplication of the periodic times. 
Also the period of III is about half the lY period, the former 
being 5-8926 days, the latter 10-9611. 

Taking only the first two decimals we find by means of con- 
tinued fractions the following approximations: 

441 1 2 3 5 28 33 

11 to 1 or z=z -, -, -, -, — -, — , etc. 

251 1 1' 2' 3' 17' 20' 

,,, ,, ,, „ 871 2 19 21 40 61 

lY " II " == -, — , — , — , — , etc. 

441 1' 9' 10' 19' 29' 

^^ ,, ^^^ , 1096 1 2 13 93 199 

V " III " — -, -, — , — , — -, etc. 

589 1' 1' 7' 50' 107' 

thus proving that only^ the fourth (Herschel II) and second 
(Lassell II) have periodic times nearly in the ratio of 2 to 1. 

The other instances adduced by Schweigger, and especially 
the first, do not seem to have any claim to be considered as real 
duplications. Still it is evident that the configuration of the 
Uranian-system is such as approaches to simple ratios between 
the periodic times ; and if the perturbating force arising here- 

^ Herschel, Outlines of Astronomy, § 550. 

^ Schweigger ; Ueber die Auffindung der ersteu Uranustrabanten durch Lassell. 
Astronomische Wachrichten. 1852, N"o. 832. 



36 G. Hinrichs on Planetology. 

from is greater than the effect of resistance, these ratios and the 
corresponding configuration would become permanent. It is 
not improbable that an analj^sis of the lunar system of Herschers 
planet will throw much light on the future configuration of the 
solar world hy ascertaining the exact relation between perturba- 
tion in commensurable revolutions brought about by resistance 
and the continued influence of the latter force on such commen- 
surable motions. 

Though this latter question cannot at present be fully answered, 
we have proved in this paragraph that not only the general in- 
commensurability of the periodic times ensuring the stability of 
the system, but also the deviations therefrom are accounted for 
by our law (38). 

§ 16. History of the Solar System. 

Believing that we have, in the preceding pages, brought forth 
some further arguments in favor of the nebular hypothesis, we 
may be permitted in a very few words to sketch the grand his- 
tory of the material universe as it is seen in the light of this 
theory. The philosophers of old called Man a Microcosmos — 
we compare the Universe, the Macrocosmos, to man, thereby in- 
timating that as Man has a parentage, growth and decay, i. e., a 
history^ so has the Macrocosmos. 

The historj^ of the raaterial world may be divided into four 
periods or ages, corresponding to those given in a note to § 6. 
(Compare Gruyot's views in Dana's Geology — chapter on ''Cos- 
mogony"). 

In the beginning God created the heavens and the earth. And 
the earth was vjithout form and void, and darkness was upon the 
face of the deep. And the spirit of God moved upon the face 
of the waters. {Genesis^ I, 1, 2). 

The material universe was created not in its present form, but 
without form ; it was void and dark ; but the spirit of God per- 
vaded it, and planned it such that his All-Foresight, or Provi- 
dence, might also be manifest in the material world. This is 
really the Creation — it is merely stated, not described, for it is 
inconceivable to mortal understanding. It is too awful, our 
mind is lost in reflecting thereon; hence the divine writer 
merely mentions it at the beginning, and, to give fullness to his 
picture and adapt himself to our understanding, describes the 
first three great ages as real creative acts, though mere conse- 
quences of the unfathomable word given in the first verse of 
Genesis. We believe that the first five verse-s of Genesis have 
never before been fully understood in their deepest sense. We 
shall in the sequel keep constantly before our eyes both this, the 
revealed History of the Cosmos^ and science^ deduced from the rev- 
elation we have in the present form of nature. 



G. Hinrichs on Flaneiology, 87 

Since Grenesis merely states that tlie universe (i. e. heaven and 
earth) was formless^ void of any organized being and dark^ it is 
science alone that can give us any idea of the constitution of the 
universe as it came from the hands of the Creator. But as 
science is progressive, our ideas of the primeval condition of the 
Cosmos must progress correspondingly, or rather with advancing 
science our eye pierces farther and farther back into the dark 
past, approaches more and more to the mysterious and almighty 
"Fiat." As these approaching steps represent greater and 
greater series of ages, we infer that th.Q Fiat lies infinitely far be- 
hind us, and can never be reached by human thought. We ex- 
perience in regard to the age of Cosmos by penetrating farther 
and farther into the dark past with our spiritual eye, the same 
that we feel in piercing, by means of more and more powerful 
telescopes, farther and farther into the world-filled abyss of 
space. Here, if looking through a giant telescope we find our- 
selves surrounded by a boundless space filled with the wonders 
of the Creator ; and if ardently searching in the existing docu- 
ments of nature for records of her past, we behold infinity also 
here, the infinity of time, eierniiy^ teeming with wonders no less 
astounding. The beautiful poem of Schiller, " die Orbsse der 
Welt,'^ is true both as to the extension and the duration of the 
World. 

The ancients most frequently thought that the world left the 
hands of the Creator in the shape it now is. Even Newton him- 
self was unable to see farther back. To him the Creator was but 
a tinker, forming his wheeling globes and wheeling them around 
their axis, putting them one by one and one by one to their 
very place in his clockwork — to him an unorganized machine to 
run on and on forever in the same shape. But Huyghens, and 
Newton himself, by discovering the generic cause of the figure 
of the earth aimed the first blow at this base idea, which never- 
theless has found its advocates even to the present hour, especially 
among theologians. The corner-stone being broken out of the 
system it has been crumbling down. Greology has restored the 
lost history of the earth, and the nebular theory has traced this 
earth to the sun as her mother. Thus creation was now identi- 
cal with the productions of the rotating mass of matter, i. e., of 
the chemical elements. 

We have attempted to show that both rotation and the ele- 
ments come from the forces wherewith the one matter (Urstoff ) 
was endowed (see § 6). It is highly interesting to see how the 
first verse of G-enesis has been understood by scientific men. It 
will at the same time more clearly set forth what we implied 
above when saying that science is approaching to the true orig- 
inal condition of Cosmos by making steps representing longer 
and longer periods of time. 
Am. Jour. Sci.— Second Series, Vql. XXXIX, No. 117.— Mat, 1865. 
36 



38 G. Hinrichs on Planetology. 

" In the beginning God created the heaven and the earth" 
means according to 

Newton, 1686: a direct, immediate creation of every globe as it 

is now. 
Huyghens, Hutton, and modern Geologists : a direct creation of 

the heavenly globes d,^ fiery masses^ circulating in the system 

as they do now. 
Kant, 1755, Laplace (later) : a direct creation of a rotating mass 

of chemical elements ; giving rise to the planetary system. 
We, in 1854, conceived this rotating mass of elements to be the 

product of a created nebula consisting of but one single element 

We will now contemplate the different ages manifest in the 
development of this Urstoff. 

First Day or Age. — The atoms of "Urstoff" combine — light 
(and heat) and the chemical elements result. The mere production 
of light would not entitle it to be considered one of the days of 
creation ; but light is by the divine writer taken as a type to 
represent itself, and the less obvious, though much more import- 
ant, chemical elements. It was not so much the light as the for- 
mation of the elements, the basis of modern physical science, 
•which characterized the first day. We think that a rotation was 
also produced hereby. (This Journal, 1864, vol. xxxvii, p. 52.) 

Second Day or Age. — Formation of the planetary orhs with their 
satellites. — The nebula developed itself into a great number of 
similar planetary nebulie, which again gave birth to similar lu- 
nar nebulae. Thus we see here the simplest kind of " life," re- 
production by division, as exhibited by many plants, and even 
animals, which to distinguish them as such from inanimate mat- 
ter, have another mode of reproduction besides. The planets 
represent the children, the moons the grandchildren of the sun. 

Third Day or Age. — The fiery balls resulting from this subdi- 
vision cool down and are shaped, as Geology has ascertained in 
relation to our own earth. 

The Fourth Age of the inorganic era is the present. We have 
shown that the further characteristic of life, namely, death, is 
not restricted to the organic but is participated in by inorganic 
nature (this Journal, [2], xxxvii, 56). As every breath of our 
lungs is a differential of decay — so every rotation of the earth 
giving us the enjoyments of another day, and every revolution 
charming us with the succession of the seasons, brings our own 
mother earth nearer to her grave/ 

^ We beg the scientific reader's pardon for these paragraphs, which do not be- 
long to this place. But we felt it urgent to say at least this much, as some, even 
to-day, are apt to base the cry of "heretic," "infidel," etc., on any such deviation 
from the beaten path in their dogmas. The nebular hypothesis has richly participa- 
ted in the abuse heaped in its day on the Copernican system, and on some leading 
doctrines of geology. Even yesterday, I found, in one of the leading religious 
quarterlies, Laplace called an " atheistic dreamer" ! We wrote this paragraph as a 
protest against such imputations. 



G. Hinrichs on Planetology. 39 

§ 17. Conclusion. 

The principal results arrived at in this paper are 

1st, A simple mechanical theory of spiral nebulas. 

2d, A more accurate determination of the orbits; and above all, 

8d, The discovery of the true law of the planetary and lunar 
distances. 

4:th, The determination of the periodic times as a function of 
the distances — or borrowing this third Law of Kepler from the 
theory of gravitation, we have therein almost a theoretical de- 
monstration of the equality of the intervals. 

As (38) what we have repeatedly called '^ our law" is very 
much like Bode's, or rather Titius's, law, we apprehend that the 
propriety of thus naming (38) will be doubted. To set this point 
in clear light we refer to a similar, though undoubtedly grander, 
case in the history of science. 

The law of Titius was exclusively derived from observation. 
It is empirical, as is the third law of Kepler. It is, moreover, 
not exact, neither in its general form nor in its numerical results. 
But neither is the famous law of Kepler exact, though, on ac- 
count of the different circumstances connected herewith, this 
latter law agrees better with the numerical data of observation 
than Titius's law. 

Newton discovered the true form of Kepler's law by deducing 
it from a higher law, that of universal gravitation. Instead of 
Kepler's form, C being the same constant for all planets, 

l^=G,. .... (70) 

Newton found that the true law is 

ju being the constant of gravitation, hence the same for all plan- 
ets; hence, 

Cx(M-j-m). . . . (72) 

That is, Kepler's constant C is proportional to the sum of the 
mass M of the sun and the mass m of the planet. By farther 
analysis it is foujid that C even is dependent on all the masses 
and distances in the system. 

So also in our case. We have given the true expression of 
Titius's law by extending it to Mercury and have accounted for 
the deviations of nature from the law, by demonstrating that it 
is a necessary consequence of the higher law, viz : the intervals 
between the abandonment of the different orbs of the same system are 
equal (see § 13). Now this is what we claim as our law. As Newton 
deduced and corrected Kepler's law by his law of equal graviia- 
tion^ so we have deduced and corrected the law of Titius by our 
law of equal intervals. 



40 G. Hinrichs on Planelology. 

We referred to (38) as our " law" because it is a consequence of 
our ]aw, and certainly our formula; we did not intend to oblit- 
erate the merit of Titius, as will be seen wherever we have men- 
tioned his name. 

There is yet another circumstance which makes our demon- 
stration of the law of planetary distances so important. It is 
the touchstone of the nebular theory ; for as this ascribes the 
formation of the planets to the slow descent of cosmical matter 
to its center, it has to be proved that such descent will give 
exactly the actual system. Already Plato held that^ "the mo- 
tion of the planets is such as if they had been all created by 
God in some region very remote from our system, and let fall 
from thence toward the sun, their falling motion being turned 
aside into a transverse one whenever they arrived at their sev- 
eral orbits." Galileo was the first who subjected this "concetto 
platonico" as he calls it to a numerical calculation based upon the 
laws of falling bodies as discovered by him. He finds an admir- 
able harmony between his calculations and the actual velocities 
and distances as they were known at his time.^ Next after him, 
Newton took the matter in hand, and in his third letter to Dr. 
Bentley he gives as his result, that it is impossible to account for 
the configuration of the system in the manner of Plato and Gal- 
ileo. This result is based upon his assumption of a vacuum. 
By taking the iDfluence of a resisting medium into account, we 
have proved that the Platonic idea as embodied in the nebular 
hypothesis does lead to the present configuration of the solar 
world. We make these remarks to show that the idea we advo- 
cate is old and venerable ; we hope, at some other time, to give 
the highly interesting history of the law of planetary distances, 
including the application of the Phyllotaxis, (Pierce, Agassiz,) 
the radius of gyration, (Kirk wood,) the regular polyhedra, 
(Kepler, Plato,) etc. 

How grand and beautiful is the harmony of the planetary 
world J What an admirable unity of plan is manifested therein ! 
As noiv the planets are slowly sinking to the sun, so they have 
always been sinking since the moment of their creation as a neb- 
ulous mass ; the same motion that now brings them nearer to 
their death has caused their formation, has brought them to life I 
And how sublime is the plan of creation ! To call forth the 
harmonious system of the solar world with all its multiform as- 
pects and dependencies fit to support life throughout almost end- 
less ages — nothing but a collection of matter endowed with its 

' Brewster, Life of Newton, Qj\y. 16. 

* Dialogo intorno ai due massimi Sistemi del Mondo, Tolemeico e Copernicano. 
Gjornata I, (ed. Opere, Firenze, 1842. Vol, i, p. 34-35.) He finds: le grandezze 
del cerchj, e le velocity del moti s'accostano tanto prossimamente a quel che ne dan- 
ne i computi, che. 6 cosa'-maravigliosa. 



G. Hinrichs on Planetology. 41 

molecular forces was placed in a little spot of the house that con- 
tains many mansions besides. This matter slowly collected to- 
gether. In thus following the force of attraction planted in it 
by eternal love, the whole great life of the solar world was 
awaked ; and as the pulsation of the heart in man indicates the 
fleeting moments of his life, so the pulsations of that great whole, 
succeeding each other at equal intervals, gave each one birth to 
a new world to mark the historic epochs of the Universe by its 
position and to roll on for ages, a revelation of the Great Au- 
thor, until, always following the same attractive force, it in death 
finds rest at the bosom of the planet-mother, the sun. And 
then — this grand system remains as a mere lump, a Cosmic Fossil, 
suspended in space, where perchance some higher being may 
meet with it, touch it, investigate it, and construct its whole past 
history, as the geologist in our days studies the history of a 
fossil shell ! 

Iowa City, Iowa, July — November, 1864. 



ERRATA. 

P. 9, line 13 from bottom, for "t= 1, 2, 3," read » f=0, 1, 2, 3." 

P. 16, foot-note 3, insert " om" before Stoffernes. 

P. 26, line 18 from bottom, in table, for " 92-3," read " 72-3." 

p. 28, in formula (47), for ^ read ^. 
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